test 3 stat ch 22
Cocaine users commonly snort the powder up the nose through a rolled‑up paper currency bill. Spain has a high rate of cocaine use, so it's not suprising that euro paper currency in Spain often shows traces of cocaine. Researchers collected 20 euro bills in each of several Spanish cities. The plus four method is particularly useful when there are no successes or no failures in the data. The study found that in Seville, all 20 of a sample of 20 euro bills had cocaine traces.
(a) What is the sample proportion 𝑝̂ p^ of contaminated bills? (Enter your answer rounded to a whole number.) Select the correct large‑sample 95% confidence interval for 𝑝p . It's not plausible that every bill in Seville has cocaine traces, as this interval says. 1 to 1
The Rails to Trails program refers to the conversion of old rail corridors into multipurpose trails for recreation and transportation. Researchers were interested in obtaining information on characteristics of users and nonusers of a 10‑mile‑long paved greenway trail in Greenville, South Carolina, that connects residential areas to both a university campus and the commercial downtown area of the city. Random digit dialing of residential numbers was done using a database of exchanges. A total of 2461 persons were contacted, of which 726 completed the survey. When a household was reached, surveyors asked to speak to the adult over 18 with the next birthday. No cell phone numbers were included in the sample. In the sample, although 726 completed the survey, only 689 of these respondents provided data on their ages. Among the 689 respondents who provided data on their ages, 253 were over 65 years old.
(a) According to U.S. census data, 13% of the population of adults in Greenville County are over 65 years old. If the 689 completed surveys that included age data are an SRS of adult Greenville County residents, do you feel that the difference between the proportion of those over 65 that completed the survey and the population proportion of 13% can be easily explained by chance variation? Select the correct hypotheses, 𝑃P‑value, and conclusion in the context of the problem. 𝐻0:𝑝=0.13 and 𝐻𝑎:𝑝≠0.13 . 𝑃≈0.0000 This is significant evidence that the sample does not represent the population of Greenville in terms of age.
The Harris Poll asked a sample of smokers, "Do you believe that smoking will probably shorten your life, or not?" Of the 1010 people in the sample, 848 said Yes.
(a) Harris called residential telephone numbers at random in an attempt to contact an SRS of smokers. Based on what you know about national sample surveys, what is likely to be the biggest weakness in the survey? The survey excludes those who have no phones or have only cell phone service. (b) We will nonetheless act as if the people interviewed are an SRS of smokers. Select the 95% confidence interval for the percent of smokers who agree that smoking will probably shorten their lives. 0.8170 to 0.8622
PTC is a substance that has a strong bitter taste for some people and is tasteless for others. The ability to taste PTC is inherited and depends on a single gene that codes for a taste receptor on the tongue. Interestingly, although the PTC molecule is not found in nature, the ability to taste it correlates strongly with the ability to taste other naturally occurring bitter substances, many of which are toxins. About 75% of Italians can taste PTC. You want to estimate the proportion of Americans with at least one Italian grandparent who can taste PTC.
(a) Starting with the 75% estimate for Italians, how large a sample must you collect in order to estimate the proportion of PTC tasters within ±0.09 with 90% confidence? (Enter your answer as a whole number.) (b) Estimate the sample size required if you made no assumptions about the value of the proportion who could taste PTC. (Enter your answer as a whole number.)
The Monterey Bay Aquarium, founded in 1984, is situated on the beautiful coast of Monterey Bay in the historic Cannery Row district. In 1985, the aquarium began a survery program that involved randomly sampling visitors as they exit for the day. The survey includes visitor demographic information, use of social media, and opinions on their aquarium visit. In 2015, the survey included 356 visitors over age 65 , of which 52 used a mobile device such as an Android phone or iPad during their visit.
(a) What is the margin of error 𝑚m , of the large‑sample 95% confidence interval for the proportion of visitors over 65 who used a mobile device during their visit? (Enter your answer rounded to four decimal places.) (b) How large a sample 𝑛n , is needed to get the common ±3 percentage point margin of error? Use the 2015 sample as a pilot study to get 𝑝∗p∗ . (Enter your answer as a whole number.)
Suppose that administrators of a large school district wish to estimate the proportion of children in the district enrolling in kindergarten who attended preschool. They took a simple random sample of children in the district who are enrolling in kindergarten. Out of 75 children sampled, 48 had attended preschool.
Construct a large-sample 95% 𝑧z‑confidence interval for 𝑝p, the proportion of all children enrolled in kindergarten who attended preschool. Give the limits of the confidence interval as decimals, precise to at least three decimal pl Construct a plus four 95% 𝑧z‑confidence interval for 𝑝p. Give the limits of the confidence interval as decimals, precise to at least three decimal places. lower limit : upper limit : lower limit :
The Centers for Disease Control and Prevention (CDC) report that gastroenteritis, or stomach flu, is the most frequently reported type of recreational water illness. Gastroenteritis is a viral or bacterial infection that spreads through contaminated food and water. Suppose that inspectors wish to determine if the proportion of public swimming pools nationwide that fail to meet disinfectant standards is different from 10.7%, which was the proportion of pools that failed the last time a comprehensive study was done, 2008. A simple random sample of 30 public swimming pools was obtained nationwide. Tests conducted on these pools revealed that 22 of the 30 pools had the required pool disinfectant levels.
Does this sample meet the requirements for conducting a one-sample 𝑧z‑test for a proportion? No, the requirements are not met because the sample has fewer than 10 successes, which violates the condition for approximating a normal distribution.
Suppose that 150 students are randomly selected from a local college campus to investigate the use of cell phones in classrooms. When asked if they are allowed to use cell phones in at least one of their classes, 40% of students responded yes. Using these results, with 95% confidence, the margin of error is 0.078
How would the margin of error change if the sample size decreased from 150 to 100 students? Assume that the proportion of students who say yes does not change significantly. As the sample size decreases, the margin of error increases.
STATE: Which type of fast‑food chain fills orders most accurately at the drive‑thru window? The Quick Service Restaurant (QSR) magazine drive‑thru study visits restaurants in the largest fast‑food chains in all 50states. Visits occurred throughout the day, starting at 11:00 A.M and ending at 2:30 A.M. During each visit, the researcher ordered a main item, a side item, and a drink and made a minor special request such as a beverage with no ice. After receiving the order, all food and drink items were checked for complete accuracy. Any food or drink item received that was not exactly as ordered resulted in the order being classified as inaccurate. Also included in the measurement of accuracy were condiments asked for, napkins, straws, and correct change. Any errors in these resulted in the order being classified as innacurate. In 2015, the ethnic category including Del Taco, El Pollo Loco, Fazoli's, Panda Express, Taco Bell, and Taco John's had the fewest inaccuracies, with only 42 of 457 orders classified as inaccurate. What proportion of orders are filled accurately in the ethnic fast‑food category? Use 95% confidence.
PLAN SOLVE: Are the conditions for the appropriate statistical procedure met? Yes, the number of successes and the number of failures are both at least 15 . Determine a 95% confidence interval for the population proportion of orders that are filled accurately in the ethnic fast‑food category. Enter the values of "𝑎a " and "𝑏b " for a confidence interval of the form (𝑎,𝑏)(a,b) . (Enter your answer rounded to four decimal places.) CONCLUDE: What is the correct conclusion in the context of the problem? We are 95% confident that between (100∗𝑎)%(and (100∗𝑏)% of orders are filled accurately in the ethnic fast‑food category.
When choosing an item from a group, researchers have shown that an important factor influencing choice is the item's location. This occurs in varied situations such as shelf positions when shopping, filling out a questionnaire, and even when choosing a preferred candidate during a presidential debate. In this experiment, five identical pairs of white socks were displayed by attaching them vertically to a blue background that was then mounted on an easel for viewing. One hundred participants from the University of Chester were used as subjects and asked to choose their preferred pairs of socks. In choice situations of this type, subjects often exhibit the "center stage effect," which is a tendency to choose the item in the center. In this experiment, 34 subjects chose the pair of socks in the center. Are these data evidence of the "center stage effect"?
STATE: Are the students choosing pairs of socks randomly? If the students were choosing socks at random, what would be the chance, 𝑝0,p0, of a pair being selected? (Enter your answer rounded to one decimal place.) PLAN: Let 𝑝p be the proportion of times the center sock is chosen. What is the correct pair of hypotheses we are to test? 𝐻0:𝑝=𝑝0 versus 𝐻𝑎:𝑝>𝑝0 SOLVE: We are able to utilize the 𝑧z statistic to test our hypotheses. Why? Select the correct response. Both the samples that are successes and failures are over 10. Select the correct 𝑧z statistic. 𝑧=3.5 CONCLUDE: Select the correct conclusion about the proportion of times the center pair of socks are selected. There is strong evidence that the proportion of times the center sock is chosen more often than all the others.
In marketing, response modeling is a method for identifying customers most likely to respond to an advertisement. Suppose that in past campaigns 81.6% of customers identified as likely respondents responded to a nationwide direct marketing campaign. After making improvements to their model, a team of marketing analysts hoped that the proportion of customers identified as likely respondents who responded to a new campaign would increase. The analysts selected a random sample of 1500 customers and found that 1248 responded to the marketing campaign. The marketing analysts want to use a one‑sample 𝑧z‑test to see if the proportion of customers who responded to the advertising campaign, 𝑝p, has increased since they updated their model. They decide to use a significance level of 𝛼=0.10
Select the correct null (𝐻0) and alternative (𝐻1) hypotheses. 𝐻0:𝑝=0.816 and 𝐻1:𝑝>0.816
Suppose that, as part of a game at a charity carnival, players are invited to spin a wheel for a chance at winning either a small, medium, or large prize. The wheel is constructed so that the probability that a player does not win a prize, 𝑝p, is 0.50. If a random sample of 40 players is selected, then 𝑝̂ p^ is the proportion of players in the sample who do not win a prize.
What is the mean of the sampling distribution of 𝑝̂ p^? What is the standard deviation of the sampling distribution of 𝑝̂ p^? Give your answer precise to three decimal places. Which of the following graphs shows the sampling distribution of 𝑝̂ p^?
A 2010 study conducted by the Health Services Research deparment of the National Center for Biotechnology Information compared mortality rates for patients with acute myocardial infarction in hospitals in states that do not have mandated minimum nurse‑to‑patient ratios with California, which does have mandated minimum nurse‑to‑patient ratios. Suppose the researchers wanted to determine if the mortality rate was higher in those states than the established rate in California. Their sampling method can be considered to be a simple random sample of a very large number of patients who were treated for acute myocardial infarction, the medical name for a heart attack.
Which inference procedure listed would be appropriate? one-sample, right-sided 𝑧z‑test for a proportion
In a Harris Poll survey, 848 smokers of a sample of 1010 smokers agreed that smoking would probably shorten their lives. Harris announces a margin of error of ± 3 percentage points for all samples of about this size. Opinion polls announce the margin of error for 95% confidence.
a) Select the actual margin of error (in percent) for the large‑sample confidence interval from this sample? 2.26% (b) The margin of error is largest when 𝑝̃ =0.5p~=0.5 . What would the margin of error (in percent) be if the sample had resulted in 𝑝̃ p~ = 0.50.5 ? (Enter your answer rounded to two decimal places.) margin of error=margin of error= (c) Select the explanation for why Harris announces a ±3 margin of error for all samples of about this size? No matter what 𝑝̂ is, the margin of error will be no more than about ±3
